How to create an air distortion/blur effect in eevee?
One way to do it would be to play with Principled BSDF Transmission and IOR.

I modeled a crude Christmas tree like shape by taking a cylinder and adding some edge loops to it. Next I selected randomly some of the faces and extruded them along individual normals. Then I moved the extruded vertices down.

and set the Principled BSDF transmission to 1 and IOR to 1.01.

This basically means that the material is almost air but not quite. It's quite close to warm air with a lot of distortion.

Remember to check Screen Space Reflections under Render Properties and Refraction under Screen Space Reflections.

and Screen Space Refraction under Material properties.
Next I animated the object to scale up, follow a curve and vanish by changing IOR from 1.01 to 1.00.